The invariant manifold structure of the spatial Hill's problem

Gómez, Gérard; Marcote, M.; Mondelo, Josep–Maria

doi:10.1080/14689360412331313039

Cited by 41 publications

(44 citation statements)

References 20 publications

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“…We only pay attention to two among the variety of families of periodic orbits of the Hill problem that are known to exist for variations of the Jacobi constant [6,7]. These are the family of 8-shaped periodic orbits, which starts from small vertical oscillations through the L 2 collinear point, and the family of Halo orbits, which bifurcates from the family of Lyapunov orbits that starts from small retrograde oscillations in the plane of the primaries about L 2 .…”

Section: Hill Problem Dynamicsmentioning

confidence: 99%

Dynamic stabilization of L2 periodic orbits using attitude-orbit coupling effects

Lara¹,

Peláez²,

Bombardelli³

et al. 2012

JAESA

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Numerical explorations show how the known periodic solutions of the Hill problem are modified in the case of the attitude-orbit coupling that may occur for large satellite structures. We focus on the case in which the elongation is the dominant satellite's characteristic and find that a rotating structure may remain with its largest dimension in a plane parallel to the plane of the primaries. In this case, the effect produced by the non-negligible physical length is dynamically equivalent to the perturbation produced by an oblate central body on a mass-point satellite. Based on this, it is demonstrated that the attitude-orbital coupling of a long enough body may change the dynamical characteristics of a periodic orbit about the collinear Lagrangian points.

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Section: Hill Problem Dynamicsmentioning

confidence: 99%

Dynamic stabilization of L2 periodic orbits using attitude-orbit coupling effects

Lara¹,

Peláez²,

Bombardelli³

et al. 2012

JAESA

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“…The two orbits With the Floquet's Theory, their linear stabilities can be evaluated; a PO is stable if all the eigenvalues of its monodromy matrix have unity magnitude (Chicone 1999). Following (Gómez et al 2005), the monodromy matrix of a PO usually has the eigenvalues in the form of {1, 1, λ 1 , 1/λ 1 , λ 2 , 1/λ 2 }, the stability index is commonly defined as s i = |λ i + 1/λ i |, i = 1, 2. The PO is linearly stable if s i < 2, while linearly unstable if s i > 2.…”

Section: Pos In the Non-averaged Modelmentioning

confidence: 99%

Modeling and analysis of periodic orbits around a contact binary asteroid

Feng

Noomen

Visser

et al. 2015

Astrophys Space Sci

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The existence and characteristics of periodic orbits (POs) in the vicinity of a contact binary asteroid are investigated with an averaged spherical harmonics model. A contact binary asteroid consists of two components connected to each other, resulting in a highly bifurcated shape. Here, it is represented by a combination of an ellipsoid and a sphere. The gravitational field of this configuration is for the first time expanded into a spherical harmonics model up to degree and order 8. Compared with the exact potential, the truncation at degree and order 4 is found to introduce an error of less than 10 % at the circumscribing sphere and less than 1 % at a distance of the double of the reference radius. The Hamiltonian taking into account harmonics up to degree and order 4 is developed. After double averaging of this Hamiltonian, the model is reduced to include zonal harmonics only and frozen orbits are obtained. The tesseral terms are found to introduce significant variations on the frozen orbits and distort the frozen situation. Applying the method of Poincaré sections, phase space structures of the single-averaged model are generated for different energy levels and rotation rates of the asteroid, from which the dynamics driven by the 4 × 4 harmonics model is identified and POs are found. It is found that the disturbing effect of the highly irregular gravitational field on orbital motion is Northwestern Polytechnical University, Xiaan, China weakened around the polar region, and also for an asteroid with a fast rotation rate. Starting with initial conditions from this averaged model, families of exact POs in the original non-averaged system are obtained employing a numerical search method and a continuation technique. Some of these POs are stable and are candidates for future missions.

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“…The problem is of three degrees of freedom, yet admitting the Jacobi constant H = −C/2. Insight regarding stability in Hill model is available through the computation of families of periodic orbits in the rotating frame [48,16,13], where information on the stability character of each periodic orbit is easily obtained [15,2,18]. Planar, retrograde periodic orbits are generally stable, and, on the contrary, corresponding planar, direct periodic orbits change to instability relatively close to the central body [48,16,17,31].…”

Section: Spatial Hill Model Dynamics 21 Hamiltonian Of the Problem Amentioning

confidence: 99%

Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter

Lara

Palacián

Russell

2010

Celest Mech Dyn Astr

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Preliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission, but which is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body problem, the spatial Hill problem is used. In some cases the validity of the averaging is limited to prohibitively small regions, thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a systems of one degree of freedom whose dynamics is analyzed. * Ephemerides section,

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The invariant manifold structure of the spatial Hill's problem

Cited by 41 publications

References 20 publications

Dynamic stabilization of L2 periodic orbits using attitude-orbit coupling effects

Dynamic stabilization of L2 periodic orbits using attitude-orbit coupling effects

Modeling and analysis of periodic orbits around a contact binary asteroid

Mission design through averaging of perturbed Keplerian systems: the paradigm of an Enceladus orbiter

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